For an action of a linear algebraic group on a scheme, the aim of geometric invariant theory (GIT) is to provide a quotient of this action (potentially on an invariant open subscheme) in the category of schemes. The theory splits into two branches, according to whether the group acting is reductive or non-reductive. The former was established in the 1960s by Mumford [1]}, building on the results of 19th Century invariant theory. The latter has been studied by several authors; in particular, the recently developed theory for non-reductive groups with so-called graded unipotent radicals has many of the favourable properties of reductive GIT [2]}. In \(§\)REF , we survey the key results of reductive GIT, including the Hilbert–Mumford criterion, and the partial desingularisation blow-up procedure [3]}; in \(§\)REF  we review the instability (HKKN) stratifications arising from work of Hesselink [4]} Kempf [5]}, Kirwan [6]} and Ness [7]}. In \(§\)REF , we summarise the key results from [2]} which enable constructions of GIT quotients for actions of linear algebraic groups with graded unipotent radicals.
